Android: Google Maps location with low battery usage

FINALLY FOUND THE SOLUTION!!!
thanks to Tristan for his answer!

By default, GoogleMap uses its on location provider, which is not the Fused Location Provider. In order to use the Fused Location Provider (which allows you to control the location accuracy and power consumption) you need to explicitely set the map location source with GoogleMap.setLocationSource() (documentation)

I am reporting here a sample activity to do that:

import com.google.android.gms.common.ConnectionResult;
import com.google.android.gms.common.api.GoogleApiClient;
import com.google.android.gms.common.api.GoogleApiClient.ConnectionCallbacks;
import com.google.android.gms.common.api.GoogleApiClient.OnConnectionFailedListener;
import com.google.android.gms.location.LocationListener;
import com.google.android.gms.location.LocationRequest;
import com.google.android.gms.location.LocationServices;
import com.google.android.gms.maps.GoogleMap;
import com.google.android.gms.maps.GoogleMap.OnMyLocationButtonClickListener;
import com.google.android.gms.maps.LocationSource;
import com.google.android.gms.maps.OnMapReadyCallback;
import com.google.android.gms.maps.SupportMapFragment;

import android.location.Location;
import android.os.Bundle;
import android.support.v4.app.FragmentActivity;
import android.view.View;
import android.widget.TextView;
import android.widget.Toast;


public class MainActivity extends FragmentActivity
    implements
        ConnectionCallbacks,
        OnConnectionFailedListener,
        LocationSource,
        LocationListener,
        OnMyLocationButtonClickListener,
        OnMapReadyCallback {

    private GoogleApiClient mGoogleApiClient;
    private TextView mMessageView;
    private OnLocationChangedListener mMapLocationListener = null;

    // location accuracy settings
    private static final LocationRequest REQUEST = LocationRequest.create()
            .setPriority(LocationRequest.PRIORITY_BALANCED_POWER_ACCURACY);

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        mMessageView = (TextView) findViewById(R.id.message_text);

        SupportMapFragment mapFragment =
                (SupportMapFragment) getSupportFragmentManager().findFragmentById(R.id.map);
        mapFragment.getMapAsync(this);

        mGoogleApiClient = new GoogleApiClient.Builder(this)
                .addApi(LocationServices.API)
                .addConnectionCallbacks(this)
                .addOnConnectionFailedListener(this)
                .build();

    }

    @Override
    protected void onResume() {
        super.onResume();
        mGoogleApiClient.connect();
    }

    @Override
    public void onPause() {
        super.onPause();
        mGoogleApiClient.disconnect();
    }

    @Override
    public void onMapReady(GoogleMap map) {
        map.setLocationSource(this);
        map.setMyLocationEnabled(true);
        map.setOnMyLocationButtonClickListener(this);
    }

    public void showMyLocation(View view) {
        if (mGoogleApiClient.isConnected()) {
            String msg = "Location = "
                    + LocationServices.FusedLocationApi.getLastLocation(mGoogleApiClient);
            Toast.makeText(getApplicationContext(), msg, Toast.LENGTH_SHORT).show();
        }
    }

    /**
     * Implementation of {@link LocationListener}.
     */
    @Override
    public void onLocationChanged(Location location) {
        mMessageView.setText("Location = " + location);
        if (mMapLocationListener != null) {
            mMapLocationListener.onLocationChanged(location);
        }
    }


    @Override
    public void onConnected(Bundle connectionHint) {
        LocationServices.FusedLocationApi.requestLocationUpdates(
                mGoogleApiClient,
                REQUEST,
                this);  // LocationListener
    }


    @Override
    public void onConnectionSuspended(int cause) {
        // Do nothing
    }

    @Override
    public void onConnectionFailed(ConnectionResult result) {
        // Do nothing
    }

    @Override
    public boolean onMyLocationButtonClick() {
        Toast.makeText(this, "MyLocation button clicked", Toast.LENGTH_SHORT).show();
        // Return false so that we don't consume the event and the default behavior still occurs
        // (the camera animates to the user's current position).
        return false;
    }

    @Override
    public void activate(OnLocationChangedListener onLocationChangedListener) {
        mMapLocationListener = onLocationChangedListener;
    }

    @Override
    public void deactivate() {
        mMapLocationListener = null;
    }
}